1 Introduction and Prerequisites 1
1.1 A Nonmathematical Introduction 1
1.2 Stationary Points and Stability(ODEs) 6
1.2.1 Trajectories and Equilibria 6
1.2.2 Deviations 7
1.2.3 Stability 9
1.2.4 Linear Stability;Duffing Equation 11
1.2.5 Degenerate Cases;Parameter Dependence 18
1.2.6 Generalizations 20
1.3 Limit Cycles 22
1.4 Waves 27
1.5 Maps 31
1.5.1 Occurrence of Maps 32
1.5.2 Stability of Fixed Points 33
1.5.3 Cellular Automata 34
1.6 Some Fundamental Numerical Methods 36
1.6.1 Newton's Method 37
1.6.2 Integration of ODEs 40
1.6.3 Calculating Eigenvalues 41
1.6.4 ODE Boundary-Value Problems 42
1.6.5 Further Tools 43
2 Basic Nonlinear Phenomena 45
2.1 A Preparatory Example 45
2.2 Elementary Definitions 48
2.3 Buckling and Oscillation of a Bean 50
2.4 Turning Points and Bifurcation Points:The Geometric View 54
2.5 Turning Points and Bifurcation Points:The Algebraic View 62
2.6 Hopf Bifurcation 68
2.7 Bifurcation of Periodic Orbits 75
2.8 Convection Described by Lorenz's Equation 78
2.9 Hopf Bifurcation and Stability 86
2.10 Generic Branching 93
2.11 Bifurcation in the Presence of Symmetry 104
3 Practical Problems 109
3.1 Readily Available Tools and Limited Results 109
3.2 Principal Tasks 110
3.3 What Else Can Happen 113
3.4 Marangoni Convection 116
3.5 The Art and Science of Parameter Study 120
4 Principles of Continuation 125
4.1 Ingredients of Predictor-Corrector Methods 126
4.2 Homotopy 127
4.3 Predictors 129
4.3.1 ODE Methods;Tangent Predictor 129
4.3.2 Polynomial Extrapolation;Secant Predictor 131
4.4 Parameterizations 133
4.4.1 Parameterization by Adding an Equation 133
4.4.2 Arclength and Pseudo Arclength 135
4.4.3 Local Parameterization 135
4.5 Correctors 137
4.6 Step Controls 141
4.7 Practical Aspects 144
5 Calculation of the Branching Behavior of Nonlinear Equations 147
5.1 Calculating Stability 147
5.2 Branching Test Functions 151
5.3 Indirect Methods for Calculating Branch Points 156
5.4 Direct Methods for Calculating Branch Points 162
5.4.1 The Branching System 163
5.4.2 An Electrical Circuit 168
5.4.3 A Family of Test Functions 171
5.4.4 Direct Versus Indirect Methods 172
5.5 Branch Switching 178
5.5.1 Constructing a Predictor via the Tangent 178
5.5.2 Predictors Based on Interpolation 182
5.5.3 Correctors with Selective Properties 184
5.5.4 Symmetry Breaking 187
5.5.5 Coupled Cell Reaction 188
5.5.6 Parameterization by Irregularity 192
5.5.7 Other Methods 193
5.6 Methods for Calculating Specific Branch Points 196
5.6.1 A Special Implementation for the Branching System 197
5.6.2 Regular Systems for Bifurcation Points 199
5.6.3 Methods for Turning Points 200
5.6.4 Methods for Hopf Bifurcation Points 201
5.6.5 Other Methods 202
5.7 Concluding Remarks 202
5.8 Two-Parameter Problems 203
6 Calculating Branching Behavior of Boundary-Value Problems 209
6.1 Enlarged Boundary-Value Problems 210
6.2 Calculation of Branch Points 218
6.3 Stepping Down for an Implementation 224
6.4 Branch Switching and Symmetry 225
6.5 Trivial Bifurcation 233
6.6 Testing Stability 237
6.7 Hopf Bifurcation in PDEs 241
6.8 Heteroclinic Orbits 245
7 Stability of Periodic Solutions 249
7.1 Periodic Solutions of Autonomous Systems 250
7.2 The Monodromy Matrix 253
7.3 The Poincaré Map 256
7.4 Mechanisms of Losing Stability 261
7.4.1 Branch Points of Periodic Solutions 262
7.4.2 Period Doubling 267
7.4.3 Bifurcation into Torus 274
7.5 Calculating the Monodromy Matrix 279
7.5.1 A Posteriori Calculation 279
7.5.2 Monodromy Matrix as a By-Product of Shooting 281
7.5.3 Numerical Aspects 282
7.6 Calculating Branching Behavior 283
7.7 Phase Locking 290
7.8 Further Examples and Phenomena 295
8 Qualitative Instruments 299
8.1 Significance 299
8.2 Construction of Nornal Forms 300
8.3 A Program Toward a Classification 303
8.4 Singularity Theory for One Scalar Equation 305
8.5 The Elementary Catastrophes 314
8.5.1 The Fold 315
8.5.2 The Cusp 315
8.5.3 The Swallowtail 316
8.6 Zeroth-Order Reaction in a CSTR 319
8.7 Center Manifolds 322
9 Chaos 327
9.1 Flows and Attractors 328
9.2 Examples of Strange Attractors 335
9.3 Routes to Chaos 338
9.3.1 Route via Torus Bifurcation 338
9.3.2 Period-Doubling Route 339
9.3.3 Intermittency 339
9.4 Phase Space Construction 340
9.5 Fractal Dimensions 342
9.6 Liapunov Exponents 346
9.6.1 Liapunov Exponents for Maps 346
9.6.2 Liapunov Exponents for ODEs 347
9.6.3 Characterization of Attractors 350
9.6.4 Computation of Liapunov Exponents 351
9.6.5 Liapunov Exponents of Time Series 353
9.7 Power Spectra 355
A．Appendices 359
A.1 Some Basic Glossary 359
A.2 Some Basic Facts from Linear Algebra 360
A.3 Some Elementary Facts from ODEs 362
A.4 Inplicit Function Theorem 364
A.5 Special Invariant Manifolds 365
A.6 Numerical Integration of ODEs 366
A.7 Symmetry Groups 368
A.8 Numerical Software and Packages 369
List of Major Examples 371
References 373
Index 395